# 37 with the initial perturbation 1 y 12 1 if 02 y 2 p

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Unformatted text preview: y) if 0=2 y < <=1 2 if 1 y1 2(1 if 1 y u = x sin 8 x cos y sin ;1 y v = ; y cos 8 sin y: Here u, v, and w are the Cartesian velocity components and = 5=3, = 6, and x and y were chosen to be small. The boundary conditions specify that u = 0 on the sides and top and v = 0 on the bottom. Solutions for the density at t = 1:8 are shown in Figure 10.3 for computations with p = 0 to 3. The mesh used for all values of p is shown in Figure 10.3. The total number of vector degrees of freedom for two-dimensional discontinuous Galerkin methods is N np. Since there are four unknowns per element ( , m, n, and e) for two-dimensional ows, there are 2016, 6048, 12096, and 20160 unknowns for degrees p = 0, 1, 2, and 3, respectively. Fluxes were evaluated using Roe's linearized ux approximation 23]. No limiting was used for this computation. A high-frequency ltering 22] was used to suppress oscillations in the vicinity of the interface separating the two uids. 10.3. Multidimensional Discontinuous Galerkin Methods 37 1/4 ρ=2 1/2 ρ=1 1/2 Figure 10.3.3: Con guration for the Rayleigh-Taylor instability of Example 10.3.1. There are solid walls on the bottom and sides and open ow at the top. The results with p = 0 show very little structure of the solution. Those with p = 1 show more-and-more detail of the ow. There is no exact solution of this problem, so it is not possible to appraise the e ects of using higher degree polynomials however, solutions with more detail are assumed to be more correct. Remacle et al. 22] also did computations using adaptive p-re nement. There is no error estimate available for the Euler equations, so they used an error indicator Ej on element j consisting of Z 3 XZ Ej = r r dV + ( j ; nb )dS : j k=1 @ jk j This can be shown 22] to be the length of the interface that separates the two uids on j . Remacle et al. 22] increased the degree on elements where Ej was above the median of all error indicators. Results using this adaptive p-re nement strategy with p ranging from 1 to 3...
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## This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.

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